Distances in 3-D

For a point $P(x,y,z)$:

• Distance from the origin $= \sqrt{x^2 + y^2 + z^2}$.
• Distance from the X-axis $= \sqrt{y^2 + z^2}$; from the Y-axis $= \sqrt{x^2 + z^2}$; from the Z-axis $= \sqrt{x^2 + y^2}$.

A useful identity the bank loves: the sum of the squares of the distances from the three axes is $(y^2+z^2)+(x^2+z^2)+(x^2+y^2) = 2(x^2+y^2+z^2)$ — exactly twice the squared distance from the origin.

The plane is written as $ax + by + cz + d = 0$, with normal vector $\vec{n} = (a, b, c)$. The perpendicular distance from a point $P(x_1, y_1, z_1)$ to the plane is

For the origin $(0,0,0)$ this collapses to $\dfrac{|d|}{\sqrt{a^2+b^2+c^2}}$. The numerator is the signed plug-in (then made positive); the denominator is $|\vec{n}|$.

Equidistant from a plane: points $P, Q$ are equidistant from $ax+by+cz+d=0$ when $|ax_P+\dots+d| = |ax_Q+\dots+d|$; dropping the modulus gives the two cases $(\text{plug}_P) = \pm(\text{plug}_Q)$.

Distance between parallel planes $ax+by+cz+d_1=0$ and $ax+by+cz+d_2=0$ (same normal):

Line through $A$ with direction $\vec{b}$; point $P$. The perpendicular distance is

Parallel lines through $A_1$ (position $\vec{a}_1$) and $A_2$ (position $\vec{a}_2$), both with direction $\vec{b}$:

Skew lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$. Shortest distance:

• The numerator is a scalar triple product (a number); the denominator is the magnitude of the cross of the two directions.
• Backwards-solve: if $d$ is given and a base coordinate is unknown, set up $\dfrac{|\text{triple product}(\,\cdot\,)|}{|\vec{b}_1\times\vec{b}_2|} = d$ and solve the resulting (often linear or quadratic) equation.
• If the lines are parallel $(\vec{b}_1 \times \vec{b}_2 = \vec{0})$ use the parallel-line formula instead.

Three recurring constructions, all producing the normal $\vec{n}$ via a cross product:

• ⊥ to two planes with normals $\vec{n}_1, \vec{n}_2$: take $\vec{n} = \vec{n}_1 \times \vec{n}_2$.
• Normal ⊥ to two lines with directions $\vec{d}_1, \vec{d}_2$: take $\vec{n} = \vec{d}_1 \times \vec{d}_2$.
• Containing two (parallel-direction or coplanar) lines: $\vec{n}$ is the cross product of the two directions (or a direction with the join vector).

Then the plane is $\vec{n}\cdot(\vec{r} - \vec{r}_0) = 0$ through the known point $\vec{r}_0$, and the distance to any point follows from $\dfrac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^2+b^2+c^2}}$.

Write the line in parameter form $(x, y, z) = (x_0 + at,\ y_0 + bt,\ z_0 + ct)$. Then:

• Crossing a coordinate plane: set the relevant coordinate to 0 (e.g. $z = 0$ for the $xy$-plane), solve for $t$, read off the point.
• Meeting a plane $\alpha x + \beta y + \gamma z = k$: substitute the parametric coordinates, solve for $t$, get the intersection point; then a 'distance' is the length from a stated point to that intersection.
• Distance measured ALONG a line (e.g. along $x = y = z$): travel from the start point along THAT line until you hit the plane — the distance is the length of that travelled segment, NOT the perpendicular distance.
• Equal-angle direction: a line making equal angles with the axes has direction $(1, 1, 1)$ (direction cosines $\tfrac{1}{\sqrt{3}}$ each).